The linear ${\text{SL}}_2(\mathbb{Z})$-action on ${\text{T}}^n$: ergodic and von Neumann algebraic aspects

The unique irreducible representation of ${\text{SL}}_2(\mathbb{R})$ on ${\mathbb{R}}^n$ induces an action, called the $\mathit{\text{linear}}\mathit{\text{action}}$, of ${\text{SL}}_2(\mathbb{Z})$ on the torus ${\text{T}}^n$ for every $n\ge 2$. For $n$ odd, it factors through ${\text{PSL}}_2(\mathbb{Z})$, so we denote by $G_n$ the group ${\text{SL}}_2(\mathbb{Z})$ for $n$ even, and ${\text{PSL}}_2(\mathbb{Z})$ for $n$ odd. We prove that the action is free and ergodic for every $n\ge 2$, that if $h\in {\text{SL}}_2(\mathbb{Z})$ is a hyperbolic element and if $n$ is even, then the action of the subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no amenable subgroup of ${\text{PSL}}_2(\mathbb{Z})$ acts ergodically on ${\text{T}}^n$. We deduce also that every ergodic sub-equivalence relation $\text{Rr}$ of the orbital equivalence relation ${\mathcal{S}}_n$ of $G_n$ on ${\text{T}}^n$ is either amenable or rigid, extending a result by Ioana for $n=2$. This result has the following corollaries: firstly, for $n\ge 2$ even, if $H$ is a maximal amenable subgroup of ${\text{SL}}_2(\mathbb{Z})$ containing an hyperbolic matrix, then the associated crossed product II${}_1$ factor $L^{\mathrm{\infty }}({\text{T}}^n)⋊H$ is a maximal Haagerup subalgebra of $L^{\mathrm{\infty }}({\text{T}}^n)⋊{\text{SL}}_2(\mathbb{Z})$; secondly , for every $n$, the fundamental group of $L^{\mathrm{\infty }}({\text{T}}^n)⋊G_n$ is trivial.